I am trying to figure out if my exiting wiring will allow me to use a 3-way switch.

I have 3-conductor wire going from a junction box to a dual-switch box. I would like to replace 1 of the existing switches with a three way switch. The other switch is wired to the same three way cable and passes power out to it's own light. The new 3-way switch would have to return the black and the red to the junction box and from there go out to yet another new 3-way switch and the light. This second new 3-way switch is not installed at all. No cable and no switch box even for it yet.

Is this something I can do or do I need a dedicated 3-conductor wire for the 3-way switch in the dual switch box?

If you can get a new 3 wire+grd cable from the existing box to the new three way location then you can do it. It does not matter about the other wiring in the switch box. You will only be dealing with the two wires that were connected to the old switch and the three new wires from the new cable.

For starters, a clear definition of terms. There are basically three boxes I am dealing with here:
1) a central junction box, from which the cables to the old switch and the new switch go.

2) the old dual switch box. This is where I am trying to replace a 2-way switch with a 3-way switch. There is 1 3-conductor + ground cable going to this box from the junction box mentioned in step 1. As i mentioned, that same cable is also used for the other switch in the box as well as the light it operates.

3) a new switch box. This will have other 3-way switch and will have a 3-conductor wire (plus ground) from the junction box mentioned in step 1.

So what I didn't understand in your reply JoeD was from which box do I need to get a 3-way conductor cable. From the dual switch box or from the junction box?

This looks like the senerio you described
This shows that you need to run a 3 sheilded plus ground from the old switch to the new switch. one will be a hot lead the other 2 are your commons. You do not need extra wires run to your light or junction box.

Well thanks for the help everyone. After thinking it over and talking with some more people it doesn't sound like what I have will work. The solution I ultimately came up with was to install some special X-10 switches which only require 1 traveller wire. They are not easy on the wallet though!